On the danger of combination resonances

ON THE DANGER OF COW3INATION RESONANCES (OB

OPASNOSTI

KOMBINATSIONNYKB

PMM Vo1.27,

No.6,

pp.1134-1142

1963,

G.

K.

REZONANSOV)

VALE&V

(Leningrad) (Zece

We have the

derived

case

of

the

simple

ived

equations

for

resonance

differential

equations

of

we investigate

of

frequencies.

introduction

duced, This

of

may result phenomenon

occur

in the

1. Let

We consider

Following

Solutions.

the

[II,

takes

Here Y is C is

meters,

(

.

0

iv (z +

24

24

= p (T),

.

of

class

piV

,U1)

solution

conjugate

friction

on the

systems

friction into

differential

‘s $

Definition

1.1.

.

. 0

.

.

.

an unstable

resonances;

introone.

it

does

not

)

p > 0 and 0 > 0 are

real

para-

N

(T)

:_

,vkp-

f Ii=-co

N,

$

(1.1)

and oS 2>0

.

E N (‘c),

-g

= (1v.JS@)(I 1’*

refer

(1.2)

/ k‘V, ( < 00

k=-cc

Ph.eikr,

(1) 1) are

We shall

we show that

equations

. . . co,,2i

in

stability

already

2

ipki<30

k=-00

kc-cc

The coefficients

in

(C $ p P (0 t)) Y = 0

vector,

.

P (t) =

on the of of

in combination

of

matrix

q2

c :=

p (T +

only

an m-dimensional a diagonal

exponents

resonance.

a system

‘2:+

effect

a stable

simple

us consider

stability

the

or an increase

place

of

the

a wider

for

friction,

in turning

case

characteristic

a linear quasistatic system of second with periodic coefficients. For a certain

class

resonant

finding

1962)

of

order

equations

19,

November

real to

1745

for the

real system

values of

of

7.

equations

(1.1)

and

1746

K.C.

(1.2)

as belonging

teristic the

to

exponents

imaginary

1.1

the

are

spect

to

the the

roots unit

by the

real Let

for

of

the

of

symmetry

condition

us indicate

by -t.

if

its

with

of

charac-

respect

characteristic

symmetry

equation

a reversible determinant

particular

of

cases

to

of

[2,

ex-

multipliers, p. 1651,

with

re-

equations

For that

(1.1)

it

is

(1.1)

is

system

does

not

system

EZ-

it

-N

E

E

to

class its

to

Class

if

t is

M.

hf.

form

N (z)

For that N* (z)

belongs

re-

let

N (-z)

reducible

of

change

to

self-conjugate.

(1.1)

systems

sufficient

P* (.c) 3 P (zl, 3) An arbitrary

mechanical process, will always [31, formed for system (1. l),

of

P(-T)GP(T), 2) System

of

characteristic

p = io and Im o = O, then

1) A system placed

systems),

symmetrically

circle.

System (1. l), describing be of class M. If the Hill’s is

mechanical located

condition

may be replaced

which

M (i.e.

p. 1681 are

axis.

In Definition ponents

class

[2,

Valeev

(t)

(1.3)

is

(1.3) sufficient

to

const

and (1.4)

let (1.4)

by the

trans-

formation

x = where the matrices and second

B(t)

for

The characteristic term

for

kei

u = 0 have

are

E

CO,

exponents

of

(k

t

fl,

= 0.

the

B (t+ 2~l6-‘) E B (t)

B (t)#0,

and B-‘(t)

derivatives

an additive which

Det

B (t) Y,

together

bounded,

with

(1.5)

their

27~0-~1. system

f2,

(1.1) are determinate UP to They fall into 2m groups

. . . ).

form (s = 1, . . ., m, k = 0, +1, _t2, . . .)

The characteristic metrically

with

continuously

exponents

respect

with

solution

of

system

t - + 0.

If

there

the is

expWI then

there

to

necessarily

the

are

of

(t) exists

axis.

change

class

an unbounded ‘pl

located

real

continuous

(1.1)

first

(Re 23

in the complex plane symThey change their position

of

parameters

CI and 8.

M cannot

be asymptotically

solution

of

>

the

form

0, cp0 + 2ne-1) = cp0)

a solution

(1.6)

of

the

form

The null

stable

for

The

‘PZ0)

Parameters

instability, ary

the

axis

the of

v and 8 reaches

equation class

of

the

For oj concerns tion

= ah we have

is

referred with

is

fulfilled

(1.7)

choice

of

sign

in

For a system

of

1.2.

the

of

coincide

of

the

region

of

the

imagin-

when the point

instability

instability

condition

the

region of of

region. of

system

multiplicity

coincidence

of of

of Hence,

(1.1) charac-

character-

form v, h = 1, . . ., m; n = 0, 1, 2, . . .)

the case

the case

of

simple

for

of

resonance;

resonance

resonance

a unique

set

when, of

if

[2, for

values

o.

p.341 1.

# oh the reThe paper

a given

of

j,

(1.7)

8,,

rela-

h and n and a

(1.7). equations v

Definition

of

to as combination

itself

1747

and p2 approach

from the condition

=e,

n-‘)oj+OhI

sonance

boundary

For p = 0 the

has the

boundary

p1

The exponents

boundary

exponents. exponents

the

exponents the

M can be obtained

teristic istic

M and 8 approach sides.

resonances

0, Im pa = Im pl, qa (t + 2~~0-9 E q1 (t))

characteristic

from opposite

parameters

combination

of

Re PI<

We pn = -

expIp,tl

When the

danger

(1.1)

of

(0) = 0

IV we always

have (1.8)

(s = 1, . . ., m)

88

A system

class

of

equations

(1.1)

in which

for

Y*Y # 0

the

quadratic

form

f (Y1t * * is

positive

If

definite if

friction, (1.9)

it is

characteristic p* (09~3 =os6 in the

first

*9

shall

belongs

Y,,,) = Y*N,Y > 0 be referred

to class

fulfilled

p, ( F,

0, k+oh # N, approximation pv

8))

as a system

of

Class

M with

N, E 0.

we have vsio)

exponent

p2 +

to

M for

(1.9)

> 0 (s = 1.

. . . . m).

For the

where (h=

we obtain

(O)p+ Co*a f (18

1,

the

. . ., m, k = * 1, f 2, . . 0)

(1.10)

equation

pa% co)+ 0 @B)= 0 88

(1.11)

For u > 0 we have

(1.12)

1748

Valeev

K-G.

Therefore, the question of stability of system (1.1) is solved only through the "resonating" exponents pi which are close to ioj for small values of 1~1 and 10 - 8,(. The region of instability of a system of class M can be found from the condition Re pj > 0. Let US note that, as a rule, SOlUtions of a system of class kf are unstable on the boundary of

the region of instability. Solutions of a system of class k! with

friction are always stable.

Definition

1.3. A frequency 8

[41, if for matrices N'(T) and P a sufficiently

will be referred to as strongly stable 9

(T),

which are arbitrary but varied to

small extent in (1.1)

I A” (T) - -v (4 I < 8,

I p’ W - P W I < e

(1.13)

(--
and which allow system (1.1) to remain in class M, the solutions of system (1.1) will be stable for all values of 0 and ~1 satisfying the condition Ie--o1<&

(1.14)

a>0

o,
where E and 6 are some positive numbers. Only a denumerable number of resonant frequencies, namely those of tbe form (l.?), may not be strongly stable. 1.4. A frequency 0, will be referred to as strongly unfor matrices N'(7) and P'(T), which are arbitrary but stable, if, varied to a sufficiently small extent in (1.1). which satisfy (1.13) Definition

and which allow the system (1.1) to remain in Class M for fj > 0,

WY

E > 0 and

the values of 8 and ~1 can be found from (1.14) such that the

solutions of system (1.1) will be unstable. In this case a wide region of instability will be adjacent to the resonant frequency 0,. bet US agree to call the constant symmetric matrix C a simple POSitive definite matrix, if a positive definite quadratic form corresponds to it. 2. Below we consider basically the combination

resonance with fre-

quencies 0, which are close to the resonant frequency (2.1)

f30=0,,j,h=n-110j+~hl The

formulas for another resonant

e,*

~0

t = n. I. h

n-ljoj-_o

freWencY

h

’

8,.

oj>Oh

(2.2)

The

danger

of

1749

resonances

combination

can be obtained from the formulas for case (2.1) if uh is replaced by ‘Oh. The frequencies 0s j ,, and 0; j h will be called conjugate fre, , , quencies. l

We look series

for

the solution

Y(t) =CP

of system (I. I)

in the form of a vector

5 Y8eaiet, 8=-m

(2.3)

Substituting Y(t) from (2.3) into (1.1) and setting the coefficients at different exponents equal to zero, we find an infinite system of 1 inear algebraic equations

[N&r (p +

iE b + k@ i)” + cl yk + p g

sei) +

pk_& y, = 0

*=-CO

(k = 0, f 1, f 2, . . J Let us change from (2.4) signet ions

to scalar

notation

!,ka = -

(2.4).

solved

system (2.6)

I CLI <

El?

@ +

the de-

SC) + Jl$-6)

(2.5)

1,2,.

(2.6)

yka, take the form

f,k,.“y,, (k=0,*1,&2,.

W, (k) 5

We consider

for

and introduce

f&S” = P-@ as llr

d, (k) = [(P 4 kOi)s + o,a)-‘, Equations

(2.4)

. .,

a=

. .,m)

in the region

10 -801
IP-

iOjI


(2.7)

It is assumed all the time that, given 8, and 8,*, relations (2.1) and (2.2) are fulfilled only for a unique set of values of j, h, n (j, h = 1, 2. . . . . II; n = 1, 2, . . .). For sufficiently small values of and ~3 all the functions d,(k) (2.5) will be bounded in region E1* Ez (2.7) except two: dj(o) and dh(-n). (2.1) and (2.5). In the latter ones the denominators become equal to zero for 8 = 6, and p = ioj. Let us exclude the two equations with the indices k = 0, a = j and k = - n, The remaining equations will form a completely a = h from system (2.6). regular system [5, p.1671 for sufficiently Small Values of EI, ~g and sj

in (2.7) m

Y ka -

-

W, V4 2

M

r)’

.fct.'**~sr

- @,(4

VA".yoj

+ f,;-"y_J

(2.8)

1750

K.C.

Here and below terms

with

solution

of

in the

n are

s = -

summation

m

co

2

2’

by the method

d, (k) f$d,

J+

(s) f,;*"-

. . . Y0j

s=--co

i-=1

$ ;’

da (k) f,ftsd,(4

i CL’

f,,“,s-”- .

1

r= 1 s=-

Substituting

(2.3)

the

omitted. The of successive

We have

/,;zo+ v2

pd, (k)

sum means that

can be obtained

1601.

P.

vd, (k) fat-”

+[-

in the

s = 0 and r = h,

(2.8)

[5,

1-

=

stroke

r = j,

system

approximations

Yki,

the

indices

Valeev

into

the

remaining

(2.9)

Y_~

two equations (2.10)

with

indices

a = j,

k = 0,

The known quantities

all

=

p*

=

yjjy

-

a21

=

Pfhj-wO -

for

(s,

k = 1,

pfjl;o’-

0j2 +

+

al2

The condition (2.11)

ask

2

p2

the

~2

(2.13)

which

be obtained

are

from

will close formula

Here and below

(2.1) and (2.2). (except for the (2.13)

(2.11)

2’

fj;-“dr

2’

fh;flfsdr(s) f,;*"

existence

of

the

form

(s) r,s;-”

frsso + 0 (P)

($

(2.12j

3_ 0 ~“)

-+ 0

(p3)

the non-zero

solution

of

system

is

Equation

3.

system

cc

5 fjf’sd,

all=22

ponents

2) have

~’

92 ~

the

a!21yoj+ %!y_nh = 0

0,

a12Y_nh =

'ilY,)j +

n and a = h we obtain

k = -

-

a12a21

permit

the

to

in region

ioj

(1.6)

we will

the values interchange

[61.

=

(2.13)

0

determination

of

(2.7).

characteristic

ex-

The same equation

The derivation

given

here

is

could simpler.

always of of

assume that for given 8, and 8,. in j, h and n can be chosen in a unique way places of oj and oh). Let us set in

and (2.12) p =

ioj +

izp,

0 =e,+

htb

00 =

n-l

(Oj

+

Ofi)

(3.1)

The

with

the

accuracy

of

danger

up to

small

combination

values

1751

resonances

Equation

O(v).

(2.12)

takes

the

form

iv.(P) + 33

n.w / II

iv,,‘j-n’ $

Equation

(3.2)

For the

resonance

ponents

differs

(J).

(n)

22

-

ivhj:) -

sCh(j-n’ / Oj

can be obtained part from

of

only

xhJIo)i Wh -

the

by the

equation sign

= 0

equation

for

at ah.

’

2hn

22 +

from a more general

Be* (2.2)

(3.2)

nj’h”’/o;

-

“,h

1

(3.2)

(5.6)

[3].

characteristic

Let

ex-

us introduce

the

designation

(3.3)

The bar denotes the

the

complex

conjugate.

Quadratic

equation

(3.2)

has

solution

(3.4)

For system

(1.1)

of

class

U without

Since

z 1 and z2 must be located

axis,

the

1x1 h = 0,

expression i.e.

the

radical

Im g = 0 (3.3).

Theorem 3.1. necessary

under

In order

for

friction

Vjj(0)

symmetrically sign

with in

(3.4)

Hence we have the system

(1.1)

to

= “h,,(O)

respect

= 0.

to the

must be real

real for

theorem.

belong

to

Class

U it

is

that

(3.5)

Theorem

possible in

Jl,ij

Conditions

region zero.

(‘I)

-=

arg

(3.5)

For a system to

for

system. (1.1)

definite.diagonal

to belong

matrices

C,

to

it

is

Class

M for

necessary

all

that

(1.1) arg

the

In order

3.2.

positive

of Hence,

n;,,

01)

~_ arg

and (3.6)

(1.1)

of

instability it

(71)

follows

class the that

_

v,,j

are

3

_;j.

fulfilled

M without expression

arg v,,,

(‘1)

-z

_

in cases friction, under

+

(mod n)

(1.3) on the

the

radical

(3.6)

and (1.4). boundary sign

is

of equal

1752

K.C.

Valeeu

(3.7)

where g is (1.1)

of

determined

class

in

CPY

x

From formula

definite

Let N(T) 5 0

where vj6-“)

For a system diagonal

1) The frequency

of

matrix

y = 0

the

case

of

a system

01>, 0)

= vji,“)

= 0,

equations

(3.8)

(3.8)

follows of

the

class

theorem.

M with

a posi-

C:

6 = 8” i h (2.1)

will

(-~qw

<

h (2.1)

will

,

2

‘hj

2) The frequency

us consider

cc+ pp (WI

-t

(3.7).

3.3.

Theorem

tive

(3.3).

M in which

8 = on j

, ,

be strongly

stable

for

0

(3.9)

be strongly

unstable

for (3.10)

Tchy%cj(~)> 0 3)

is

4) is

The frequency

8 = 6;

The frequency

will

be strongly

stable

8 = tf’, I h (2.2)

will

be strongly

unstable

The last

two statements

conjugate

frequence

sign. System

(3.8)

with

with

Hence,

from Theorem 3.3

Theorem

(2.2)

a diagonal

3.4.

If

definite is

of

8;

j

I J

the

for matrix

unstable;

strongly stable, and vice versa,

strongly (provided

unstable, that the

theorem

h, the

a positive

system

positive

, ,

, a

fulfilled.

the

j

h (2.2)

fulfilled.

definite

matrix

follow

expression

matrix

C by a linear

follows a system

the of

C on% of

3.1.

(3.9)

that

changes

C can be reduced

transformation

equations the

(3.8)

frequencies

of 8,

then its conjugate frequency will assumptions made at the beginning

of

if

fact

g (3.3)

of

for the

to

form

class

W with

and 8,.

(2.1)

then its conjugate frequency if one of the frequencies 6,

For a system

(3.10)

the (1.5).

theorem.

fulfilled). Example

from the for

if

two equations

n and

will be strongly and 8,. is

be strongly stable of the section are

The

qyl+

-+

dt2

@+

by comparison

of combination

danger

with

From Theorem

(1.1)

and (1.2)

3.3

(1) = -

%l

it

follows

or= q strongly

stable,

the

only

strongly

For a canonical P’(T)

Theorem

definite

(2.1)

cannot

cannot

8,

cannot

Statement

z

n.

jh

ZQ) the

= -

1,

Jcpl@) =

2

(3.12)

frequencies

8, = 0.5 (WI fi 01,)

(3.13)

frequencies e,* = 0.5 (oi-

%

(1)*)

(3.14)

ones. differential

(*),

equations

(3.8),

where P(T)

=

(2.1)

g =

.hj(-n)njh(*)

theorem

in a system

is

(3.8)

zz

1 fijh@) 12>

(3.15)

0

valid.

of

class

M with

a diagonal

positive

C:

P(T) is symmetric be strongly

be strongly

2) The matrix

P(T)

= P*(T), then the frequencies and their conjugate frequencies 8,*

P(T)

stable.

unstable. is

cannot

be strongly 1 follows

skew-symmetric be strongly

P(T) = unstable,

then

P*(T),

and the

the

frequencies

fre0,*

stable. from

a theorem

due to

Krein

[Z.

p.3531

and [4.

4931. Note that

8,

of

ol>on

i-5 t

02,

1-

following

If

3.5.

matrix

quencies

p.

0

system

the

1) The matrix

(2.2)

(3.11)

0 we have

Therefore,

(2.2)

9

unstable

fihj@)

8,

2 cos 29 t) yi = 0

we find

that

and the

e,* = are

1753

+o,~y,+p(-3coset+4cos2et)yl=0,

nla(i’ = 0.5,

are

p (cosf3t -

resonances

to

if

be strongly

matrices

Pi

unstable,

and P2(~) the total

in

(3.8)

perturbation

cause matrix

some frequency

P(T) (3.16)

may make the

same frequency

strongly

stable.

1754

Re

K.C.

are assuming

QT),

Pz(??

Example frequency

that

and P(T) 3.2.

afX the

(3. It;)

For a system

of

= o1 + wz will P2(-r) and f)(7)

For a System frequency

8,

(3.8)

with

a system

Tbe following

the

of

total

(3.6)

with

equations

(3.6)

unstable

for

perturbation

GlElsS

are

equations

w,

matrices

c

two differential

be strongly

(I,11

theorems

of

class

be strongly

= w1 + w2 will

Consider

systems

are of

6,

matrices

Voleev

the

matrix

the

perturbation

f’(r)

(3.16)

the

stable. if

P(T)

the consequence

= 6

of

formulas

(3.7)

of

I with

and

(3.3). Theorem

diagonal

3.5,

a system

For

positive

definite

1) The frequencies

on ~ ), (2.1)

and 6;

J,

7.

positive

and 8*n j

h are

strong&X~stable, strongly Theorem

3

(2.2)

!#ill

be strongly

I h (2.2)

will

be strongly

> 3

For a System of equations (3.18) of definite matrix C, both conjugate

class M with frequencies

either Simultaneously strongly stable, or ar are simultaneously neither strongly

3.8.

the

an 8

simultane~& stable, nor

un-

‘8 h

If

and

the

@*,

M(r) > t

matrix

0 n 1 h (2.1)

# I

matrix

C in a system

(3.16)

of

class

M be

definite.

the matrix (2.1)

quencies

Let

and positive

ff

‘n,j,h

2)

class

unstable.

diagonal I)

II ,

for

Theorem

(3. Is)

C: and 6* n,1,h

, I

2) The frequencies

arbitrary

equations

fin J b (2.1)

stab1 e for

stable

of

matrix

J

h

is (2.2)

“1(~) is

Symmetric IV* = W, then alI the cannot be Strongly Stable.

frequencies

skew-symmetric N* = - IV, then (2.2) cannot be strongly

all the freunStable.

and 6z,j,h

The danger

The proof matrix

of

N(T)

the

theorem

condition

(3.20)

matrix condition (3.19) is the case v .(“)v~:-~) = 0. Ih Example

of

follows is

of

1755

resonances

from the

fulfilled,

fulfilled,

For a system

3.3.

combination

fact

that

and for

including

differential

for

the

equality

TV(6 sin8 t f

by comparison

with

v12(l) zz

-

(1.2)

8 sin 28 t) dy, / dt $

in

0

(3.21)

(a~>

= 0

~?y,

we find

via(2) = 2i,

i,

sign

equations

Gy,/dt2f~(2sin6t-4sin26t)dy,/dt~o&=O d2y2 I dt2 f

a symmetric

a skew-symmetric

v,,(l) = -

va1(2) = -

3i,

4i

(3.22)

The frequencies 91 =oi will

be strongly

stable,

el*

+o,,

and the

=01-o

frequencies

e,* = 0.5 (ml will

be strongly A similar

quencies that

the

acting

not

relation exist

(3.3)

is

4. Let friction.

both

of

in the

perturbations,

separately,

may cause

which

cause

effects

stability

are

neither

for

us consider

the

The inequality

can be transformed

the

JJ

stability

I

fi.w

-?.-Oj

N (7)

stability

together.

the

a,

+

the

form of

which,

instability, quantity

g

(1.5).

class

M with

b and c

ib>c>O

(4.1)

b2 > 42 (9 + a)

(4.2)

of if

equations

(1.1)

may be unstable

a

+ ?!!J

strong

freNote

and PC-~),

nor

equations

real

(1.1).

Obviously, of

of

conjugate

a system

the form

Solutions of a System > 0 and v,,;‘) > 0,

Im

(3.24)

mutually

of

matrices

transformations

containing

into

case

strong

when acting

an invariant

for

general

Imva+

V.(O)

aa)

unstable.

simple

does

(3.23)

a

i (vjy) -

Corresponding to inequality solved for h becomes

v$)) (4.3))

-

2hn

1

for

+vhf)) >0 1 >(v.‘!’ ‘I.

-

4g

an inequality

(4.3)

33

of

the

form

(4.2).

1756

where g is

defined

by formula

(3.3).

The value

of

the

expression

under

the radical sign in (4.5) for arbitrarily small values of vjio) > 0, (0) > 0 may be arbitrarily large. The expansion of the region of in“hh stability may take place unly for vii (0) f vhh(O) and j f h, i.e. in the case of combination resonance. The expansion itself occurs on account of

a change This

in the

interesting

properties those If

of

between

phenomenon

were demonstrated class

i.e.

if

stable,

introduced.

From (3.4) and vhh(0)

v jj (O)

was first

and vhLo). noticed

on a system

of

in

[11 where

equations

its

basic

much simpler

than

M.

g < 0,

strongly is

relation

the

they Let

solut,ions

will

g :’ 0,

we find

the

remain i.e.

of

let

expansion

system

strongly the of

(1. 1) without

stable

frequency

quantities

Im z1,1 = 1/1 [vi!;“) + v,,$’ + a (aa -- 4&11 (vjy) -

(4.5) 8,

friction

are

when friction

be strongly

Im 11 2 for

unstable.

small

v .(.O) JJ

v$,@)] + 0 ( 1vj$@/ * + 1v$) 13)(4.6)

where

In the for “hh

region

of

stability

of

system

(1. 1) without

friction,

i.e.

o2 - 4g > 0 when g .’ (I one can always choose the numbers vjjo), (0) in such a way that the coefficient at the imaginary part of one

of the stable.

roots

Comparing Theorem

t1,2

will

formulas 4.1.

If

be negative.

(I. 6)

%I +

I&

+

and (3.7’)

in a system

diagonal matrix C a certain of instability adjacent to 0 bL2)

Then the

+

we arrive

(1. 1) of

frequency it p’h, +

solutions

6,

0 @?,

then upon introduction of friction vjlo) aries 0, 2 of the region of instability

class

will

at the M with

h (2.1)

h,,X,

const,

=

plane

of

> 0,

definite

AZ # 0 the

un-

theorem.

has a wide

J., -

> 0 and vhko) in the

following

a positive

= 6” j

, ,

become

range

(4.8)

bound-

parameters

p, 6

The

takes

the

of

If

4.2.

definite

Theorem

shows

4.2

relatively

Let

region.

in the

plane

v.

the

where 8,(O)

special

into

8 for

is

will

expression

the

the

a system

boundaries = e”

solve

be real

under

the

the

of

of

arises region

for

form of

equation

- pO)p-’

If

5.1.

resonant

conditions positive,

sign

for h. which

in a system

frequency

of

Section

then

this

3 are strongly

in

8,

of or

fulfilled) unstable

the

region

resonances,

of

since

the boundaries of

the

region

the of of

instability

integer

g > 0,

insta-

by the of

o,(l~)

method con-

and 8,(p), powers

order

under the equations

If

the

of

found

higher

being

of

possibility

instability

series

was not

[71.

M are

are

of

~1.

re-

radical sign will of class hf this

then,

setting

the

(3.7) equal to zero, one can always We obtain two real expressions for are

analytic

equations

eo*

the

frequency in

about

1x1 E = Im A = 0.

radical

resulting

Theorem

a

present.

class of

then

of

combination

instance of

(1.1)

the

unstable,

combination

boundaries

in the

A, i.e. for (e,,,(cl) have the theorem.

certain

M with

CI > 0.

If in equation (3.2) the infinitesimal terms tained, then in formula (3.7) the expression contain an additional function O(p). For the function

class

a widening

of of

analyticity

that

to

always

as for

The question

= 8,(O)

of

strongly

values

danger

past

the

is lead

small

recent

parameter.

structing

frequency

friction

Suppose

(1.1)

strongly stable, then the introduction will keep it strongly stable.

consideration,

us look

bility small

the

a small

sufficient

equations

can always

sufficiently

systems

In the

of

friction

for

of

is

combination

of

instability

5.

1757

resonances

C:

frequency 8, small friction

2) A certain introduction

given

combination

in a system matrix

1) A certain sufficiently

in real

of

form

Theorem

positive

danger

(2.1) the

(1.1)

for

of

and (2.2) expression

frequency

will

l_~= 0.

class

Hence,

M for

(assuming for

a

that

g (3.3)

border

we

the

is

on a region

of instability with boundaries O,(u) and 8&Q. Here 8,(p) and 8,(p) are analytic functions of v in the sufficiently small neighborhood of the point

~1 = 0.

one, 5.1. The condition g > 0 is in a way a necessary from the following example. With additional restrictions the requirement upon a system (1.1) of class M (for instance, Note

seen

as can be imposed that the

K.C.

1758

equations the

be canonical)

analyticity Example

of

the

0,(p)

condition

g > 0 will only be sufficient point cc = o.

and O,(p)

Consider

5.1.

Valeev

a system

of

differential

dzyl/dta + 01~~1 + 2alyz cos I_%= 0,

Expressions

(2.12)

Setting

f3i)2 -+ Co22-

p =

al-O,

accuracy

up to

iol

+

ala2 [(p -

2@i)* +

and 8 = 6,

ir

+ A,

(5.1)

a,-0

O(c~l~a2~)

have

a,az I(p + 0i)2 d- 0221-1-+ 0 (a12a,2),

ulr = p2 -+ 0 12 a2$ = (p -

with

equations

d”yz,‘dt” + (o2”1/2$m l!xgyl cos Bt = 0

O,=o,+o,,

6) I >O 2,

for

at the

022)1-1 +

we obtain

the

al2

form

from

(5.2)

= a1

0 (a,*at2),

azl = a,

(2.13) (5.3)

If

al

aries

= pal,

(5.3)

a2 =

for

v2a2,

ala2

u > 0 take

>

the

then

0,

the

equations

of

the

bound-

form (5.4)

In this aries

have

case

g = 0.

an algebraic

In conclusion,

let

treated

in a fashion

ary

a canonical

for

the

expressions

singular us note

that

analogous

to

system

6,(p)

point

and 8,(p)

for

some introductory [81,

for

the

bound-

~1 = 0.

and the

was investigated

in

questions

analyticity

were

of

the

bound-

[91.

BIBLIOGRAPHY 1.

Schmidt,

G. and Weidenhammer.

linearer 2.

Malkin,

Swingungen. I. G.,

problems tizdat, 3.

Valeev,

of

Math.

Nekotorye the

zadachi

theory

of

Instabilitkten

F.,

Vol.

Nachrichten,

teorii

nonlinear

gedlimpfter 23,

nelineinykh oscillations).

Nos.

4-5,

kolebanii

rheo1961. (Some

Gostekhteore-

1958. K. G.,

‘K

metodu Khilla

v teorii

lineinykh

differentsial’nykh

uravnenii s periodicheskimi koeffitsientami (On the Hill’s method in the theory of linear differential equations with periodic coefficients). 4.

Krein,

M. G.,

PMM Vol.

Osnovnye

24.

No. 6.

polozheniia

1960. teorii

A zon ustoicbivosti

nicheskoi sistemy lineinykh differentsial’nykh periodicheskimi koeffitsientami (Fundamentals

uravnenii s of the theory

kanoof

A

The

regions

of

equations

5.

spaces). 6.

Valeev,

and Akilov.

8.

0 krutil’nykh

oscillations

No. 9.

5,

concerning equations

with

Androno-

anal iz

v normi-

periodic

rabotam

on the

theory

No.

2,

valov

1961. (On the

1,

1934.

po sistemam

systems

coefficients).

of

Deter-

2, No.

s periodicheskimi

some works with

25.

PMMVol.

k nekotorym

kharak-

in the

coefficients.

kolenchatykb

crankshafts).

uravnenii

method

PMMVol.

kolebaniiakh of

differentsial’nykh Opredelenie

periodic

expcasnts).

linei-

koeffitsienof

linear

PM! Vol.

21,

1957.

Iakubovich, V. A., 0 dinamicheskoi the dynamic stability of elastic Vol.

differential

i A .A.

in normalized

lineinykh

(On the Hill’s

V. A. , Zamechaniia

nykh differentsial’nykh (Remarks

nyi

Analysis

koeffitsientami.

equations

N. E.,

differential

linear

‘pariat

Funkt sional’

v teorii

pokazatelei characteristic

Iakubovich, tami

G.P., (Functional

K metodu Khilla

of

torsional

of

1955.

s periodicheskimi

mination

1759

1959.

differential

Kochin,

Sb.

A.V.

teristicheskikh linear

system

coefficients).

prostranstvakh

K. G.,

resonances

a canonical

Nauk SSSR,

Fizmatgiz,

uravnenii

7.

of

periodic

Akad.

Kantorovicb, rovannykh

combination

of

stability with

Izd.

va”.

danger

121,

No.

4.

ustoichivosti systems).

uprugikh Dokl.

Akad.

sistem Nauk

(On SSSR,

1958.

Translated

by 0. S.